Optimal Shapes Maximizing the Steklov Eigenvalues

Source abstract

In this paper we consider the problem of maximizing the $k$th Steklov eigenvalue of the Laplacian (or a more general spectral functional), among all sets of ${\mathbb R}^d$ of prescribed volume. We prove existence of an optimal set and get some qualitative properties of the solutions in a relaxed setting. In particular, in ${\mathbb R}^2$, we prove that the optimal set consists in the union of at most $k$ disjoint Jordan domains with finite perimeter. A key point of our analysis is played by an isodiametric control of the Stelkov spectrum. We also perform some numerical experiments and exhibit the optimal shapes maximizing the $k$th eigenvalues under area constraint in ${\mathbb R}^2$ for $k=1, \dots,10$.